Some series equivalent to the extended Riemann hypothesis for Dedekind zeta functions
Vincent Nguyen
TL;DR
This work addresses the extended Riemann hypothesis (ERH) for Dedekind zeta functions by deriving a closed-form identity over the non-trivial zeros that is equivalent to ERH. The authors define an auxiliary function $X_K(s)$ built from the xi_K function and use Hadamard factorization to express derivative relations in terms of the zero set, establishing the equivalence: ERH for $zeta_K$ ⇔ sum over non-trivial zeros of $1/|1/2 - rho_K|^2$ equals $X''_K(1/2)/X_K(1/2)$. The converse direction shows that the identity forces all non-trivial zeros to lie on the critical line, with the case $K=Q$ recovering Suman-Das. Overall, the result extends a known zeta-zero equivalence to general number fields, enriching the landscape of ERH-related criteria and providing a new tool for analyzing zero distributions.
Abstract
The extended Riemann hypothesis (ERH) for Dedekind zeta functions remains one of the most elusive open problems in number theory. Over the last century, many equivalent statements to the classical Riemann hypothesis alone have been discovered. We prove that the closed form of some infinite series over the non-trivial zeros of Dedekind zeta functions holds if and only if ERH is true.